I don't have one exact favorite anime, but I'll break it down into genres for you.
Comedy: Jungle wa Itsumo Hare nochi Guu, Excel Saga
These anime are so hilariously funny and off the wall, I actually laughed out loud. Excel Saga is so random, although it does lose it's authenticity around the later episodes (but the last one is, by far, the funniest). Hare nochi Guu's torment of Hale is fully funny, and the characters all have some strange quirk that somehow ruins Hale's life.
Action: Noir
Loads of style, nice animation, nice music, nice story, nice characters. Nice, etc.
Drama: Neon Genesis Evangelion(?)
Although I didn't like the mecha too much, the storyline is pretty good. The ending was a little disappointing, but the movies make up for it, I guess.
Romance: Ai Yori Aoshi
VERY nice animation and graphics, fan service, sweet story. Actually, fan service can be counted as a bad thing, but... Meh.
Horror/Suspense: N/A
I have yet to see one of these, maybe I'll watch Hellsing.
Mystery: Perfect Blue
This was a pretty cool movie, the ending was especially good/fascinating. Pretty old, but still very good, technically.
Other: .hack//SIGN, .hack//dusk
Very original, though being milked for most of what it's worth. :/ Many people have already seen it, so there's no point for a mini-summary.